Storage of Powders and Bulk Solids in Silos

Dietmar Schulze

Problems with the storage of bulk solids in bins and silos can be avoided
if they are designed with respect to the flow properties of the bulk solid
which has to be stored. The following essay considers the basic rules for
the design of silos.

1 Stresses in silos

Figure 1 shows silos and the pressures and stresses, respectively, acting
in the silos. While the pressure (for fluids we will use the word
pressure") would increase linearly downwards if the silo would have
been filled with a fluid (a), the course of the vertical stress (for bulk
solids we will use the word stress") in a silo filled with a bulk solid
is rather different (b,c): In the latter case in the vertical (cylindrical)
section of the silo the vertical stress increases in a degressive way. If
the height to diameter ratio of the silo is sufficiently large (usually:
> 3), a constant vertical stress is attained. This means that the vertical
stress will not increase further even if the filling height is much larger.
The reason for this course are the shear stresses acting between the bulk
solid and the silo walls even if the bulk solid is at rest. Due to the shear
stresses, the silo walls carry a part of the weight of the bulk solid. A
method for the calculation of the stress course in the vertical section was
derived by Janssen already in 1895 [1]. This method is the basis for most
present standards for the calculation of the load on silo walls for structural
silo design [2-4].

Figure 1: a. pressure in a silo filled with a fluid (imaginary); b. vertical
stress after filling the silo with a bulk solid; c. vertical stress after
the discharge of some bulk solid

The stresses acting in a hopper are different from those in the vertical
section. Just after filling an empty silo, the so called filling stress state
(also: active stress state, figure 1b) prevails, where the vertical stress
in the hopper decreases less in the upper part of the hopper and then more
near the imaginary hopper apex. As soon as some bulk solid is discharged
for the first time after filling, the stresses in the hopper change and the
so-called emptying stress state (also : passive stress state) prevails, figure
1c. When flowing downwards in the hopper, the bulk solid is compressed in
the horizontal direction so that the walls of the hopper carry a larger part
of the weight of the bulk solid and, hence, the vertical stress in the lower
part of the hopper is clearly smaller than after filling. In the emptying
stress state the vertical stresses in the lower part of the hopper are nearly
proportional to the distance to the imaginary hopper tip or, in other words,
the stresses are proportional to the local hopper diameter. This linear course
of stress is called the radial stress field [7]. In principle, in the vertical
section of the silo the stresses remain unchanged at discharge.

2 Flow Profiles: Mass Flow and Funnel Flow

Two different modes of flow can be observed if a bulk solid is discharged
from a silo: mass flow and funnel flow (figure 2a). In case of mass flow,
the whole contents of the silo are in motion at discharge. Mass flow is only
possible, if the hopper walls are sufficiently steep and/or smooth, and the
bulk solid is discharged across the whole outlet opening. If a hopper wall
is too flat or too rough, funnel flow will appear. In case of funnel flow
(figure 2b), only that bulk solid is in motion first, which is placed in
the area more or less above the outlet. The bulk solid adjacent to the hopper
walls remains at rest and is called dead" or stagnant" zone.
This bulk solid can be discharged only when the silo is emptied completely.
The dead zones can reach the surface of the bulk solid filling so that funnel
flow becomes obviously when observing the surface. It is possible as well
that the dead zones are located only in the lower part of the silo so that
funnel flow cannot be recognised by observing the surface of the silo filling.

Figure 2a: Mass flow

Figure 2b: Funnel flow

3 Flow Problems

Typical problems which occur at the storage of bulk solids are:

Arching: If a stable arch is formed above the outlet so that the flow
of the bulk solid is stopped, then this situation is called arching (figure
3a). In case of fine grained, cohesive bulk solid, the reason of arching
is the strength (unconfined yield strength) of the bulk solid which is caused
by the adhesion forces acting between the particles. In case of coarse grained
bulk solid, arching is caused by blocking of single particles. Arching can
be prevented by sufficiently large outlets.

Figure 3a: Arching

Ratholing occurs in case of funnel flow if only the bulk solid above
the outlet is flowing out, and the remaining bulk solid - the dead zones
- keeps on its place and forms the rathole. The reason for this is the strength
(unconfined yield strength) of the bulk solid. If the bulk solid consolidates
increasingly with increasing period of storage at rest, the risk of ratholing
increases. If a funnel flow silo is not emptied completely in sufficiently
small regular time intervals, the period of storage at rest can become very
large thus causing a strong time consolidation.

Figure 3b: Ratholing

Irregular flow occurs if arches and ratholes are formed and collapse
alternately. Thereby fine grained bulk solids can become fluidized when falling
downwards to the outlet opening, so that they flow out of the silo like a
fluid. This behaviour is called flooding. Flooding can cause a lot of dust,
a continuous discharge becomes impossible.

Wide residence time distribution: If dead zones are formed (funnel
flow), the bulk solid in this zones is discharged only at the complete emptying
of the silo, whereas bulk solid, which is filled in later, but located closer
to the axis of the silo, is discharged earlier. Because of that, a wide
distribution of residence time appears which is disadvantageous in some cases
(e.g. in case of storage of food or other products changing their properties
with time).

Segregation: If a heap is formed on the bulk solids' surface at filling of the silo, segregation is possible according to particle size or particle density (figure 3c). In case of centric filling as shown in figure 3c, the larger particles accumulate close to the silo walls, while the smaller particles collect in the centre. In case of funnel flow, the finer particles, which are placed close to the centre, are discharged first while the coarser particles are discharged at the end. If such a silo is used, for example, as a buffer for a packing machine, this behaviour will yield to different particle size distributions in each packing. In case of a mass flow, the bulk solid will segregate at filling in the same manner, but it will become "remixed" when flowing downwards in the hopper. Therewith, at mass flow the segregation effect described above is reduced significantly. (A short movie showing segregation due to particle size you will find here).

Figure 3c: Segregation

In a funnel flow silo, all problems mentioned above can occur generally,
while in case of mass flow only arching has to be considered: segregation,
ratholing, irregular flow and flooding of the bulk solid do not appear in
a well designed mass flow silo. The residence time distribution of a mass
flow silo is narrow, because it acts as a first in - first out" system
(see figure 2a).

Two steps are necessary for the design of mass flow silos: The calculation
of the required hopper slope which ensures mass flow, and the determination
of the minimum outlet size to prevent arching.

4 Silo design

The flow behaviour of a bulk solid is defined by several well-defined parameters
[2,5-8,21]. In general, these are the bulk density rb, the effective angle
of internal friction je (a measure
for the internal friction of the bulk solid at stationary flow), the unconfined
yield strength sc, and
the wall friction angle jx. For mass flow design,
the wall friction angle jx is the most important
parameter, whereby the unconfined yield strength
sc is the most important parameter
regarding arching. The wall friction angle jx is defined as the friction
angle between the surface of the silo wall and the corresponding bulk solid.
The unconfined yield strength sc is the compressive strength
of a bulk solid. It has to be taken into account that all these parameters
are dependent on the stress level, represented by the consolidation stress
s1 [2,5-8,21].

The parameters mentioned are measured in dependency on the consolidation
stress with shear testers [2,5-8,21], e.g. with the Jenike shear tester or
a ring shear tester. The hopper slope required for mass flow and the minimum
outlet size to prevent arching can be calculated with the measured values
using Jenikes' theory [7]. This method showed its validity in many cases
in more than 35 years.

The borders between funnel and mass flow, which result from the calculations
of Jenike [7], are shown in figure 4a for the wedge shaped hopper and in
figure 4b for the conical hopper. In the diagrams the wall friction angle
jx is drawn over the
hopper slope angle Q measured against
the vertical. The effective angle of internal friction je, which is a measure of
the internal friction of the bulk solid, is the parameter of the mass flow/funnel
flow borderlines. The borderlines separate all pairs of values leading to
mass flow from those leading to funnel flow.

Figure 4a: Design diagram for mass flow (wedge-shaped hopper)

Figure 4b: Design diagram for mass flow (conical hopper)

Conditions which lie within the borderline yield mass flow whereas funnel
flow is present in case of conditions outside of the borderline. If the wall
friction angle jx and
the effective angle of internal friction
je are known (measured with
a shear tester, e.g. with the ring shear tester), the maximum slope angle
Q of the hopper wall against the vertical
which ensures mass flow can be determined with this diagram. The courses
of the borderlines indicate, that the larger the wall friction angle jx is, the steeper (smaller
Q) the hopper has to be for mass
flow. The wedge shaped hopper allows a somewhat (often 8° to 10°)
larger slope angle Q against the vertical
with the same material properties. That means that the walls of a wedge shaped
mass flow hopper can be flatter than the walls of a conical mass flow hopper
[7,12].

When bulk solid is discharged from a mass flow silo, the radial stress field
prevails in the hopper as already described in section 1 (see figure 1c).
In the hopper (at least beneath a sufficiently large distance from the vertical
section) the major principal stress s1 is proportional to the
local hopper diameter (figure 5). It decreases to zero towards the imaginary
hopper apex. The stress s1 acts as a consolidation
stress thus determining the properties of the bulk solid, e.g. the bulk density
rb and the unconfined
yield strength sc. The
unconfined yield strength sc of a bulk solid can be
measured for each major principal stress (consolidation stress) s1 (see [21]). The function
sc
=f(s1) (figure 6)
is called the flow function. Usually, the unconfined yield strength increases
with the consolidation stress. If the flow function has been measured, the
unconfined yield strength sc can be drawn in figure
5 at each position of the hopper.

Figure 5: Stress conditions in the hopper (emptying)

s1' is the bearing stress
acting where an imaginary stable arch of bulk solid is carried by the hopper
walls. s1' is proportional
to the local hopper diameter such as s1. An arch can only be
stable in that are of the hopper where the unconfined yield strength sc is larger than the bearing
stress s1' . This is the
case beneath the point of intersection of the sc curve with the s1' curve. Above
that intersection the unconfined yield strength is smaller than the bearing
stress of the arch. In this case, the unconfined yield strength is not large
enough to support an arch, i.e. an arch would not be stable at this position.
The point of intersection indicates the lowest possible position in the hopper
(height h*, figure 5) for an outlet opening large enough to avoid arching.
The diameter of this minimum outlet opening is called dcrit. If
a smaller outlet opening would be chosen, h* indicates up to where flow promoting
devices have to be installed beginning at the outlet.

Figure 6: Flow function and time flow function

Some bulk solids tend to consolidate increasingly with the period of storage
at rest (time consolidation [8,21]). It can be found a time flow function
sct =
f(s1) (figure 6)
for each storage time analogously to the flow function. If the time flow
function would be drawn in figure 5 then this would yield to a point of
intersection of the s1'-curve and the
sct-curve, which would
be above the already determined point of intersection of the s1c- and s1'-curves. This
means that larger outlets are required to prevent arching with increasing
storage time at rest.

For practical silo design, equations or diagrams derived by Jenike [7] are
used to determine the stresses
s1 and
s1' in dependence on
the flow properties measured(je,
jx, rb) and the silo geometry
(Q). With this means the minimum outlet
sizes of conical as well as wedge-shaped hoppers can be calculated. Furthermore,
the minimum outlet sizes for avoidung ratholing at funnel flow can be determined
[7].

5 Choice of the hopper geometry

Figure 7 [7,9,12] shows some opportunities to design mass flow silos. The
calculations of Jenike (see design diagrams, figure 4) refer to conical hoppers
(a) and wedge shaped hoppers (b). In case of these basic hopper forms, the
maximum slope angles of the walls to achieve mass flow(Qax in case of a conical,
Qeb in case of a
wedge-shaped hopper) and the outlet dimensions (d, b) to prevent arching
can be determined. In case of the wedge shaped hopper it is assumed, that
the influence of the vertical end walls can be neglected if the length of
the outlet L is at least three times the width b.

Figure 7: Hopper forms [9]

The variants c and d are advantageous as well to ensure mass flow if the
maximum slope angles as indicated in the figure are not exceeded. The pyramidal
hopper geometry (e) is disadvantageous because the bulk solid has to flow
from the top in the edges of the hopper and in the edges to the outlet. Thus,
the bulk solid has to overcome wall friction at two sides what supports the
formation of dead zones. If mass flow has to be achieved with such a hopper
geometry, the edges have to be rounded on the inside, and the maximum slope
angle against the vertical of the edges must not exceed Qax. Because the walls of
a pyramidal hopper are always steeper than the adjacent edges, a pyramidal
mass flow hopper is steeper than a conical hopper for a specific bulk solid.
Variant f is just a transition from a cylindrical section to a square outlet.
In this case, the slope of the hopper walls against the vertical must not
exceed Qax at any position.

In order to achieve mass flow, variants e and f must have the steepest walls.
The conical hopper (a) can be designed more shallow, and the largest slope
angles measured against the vertical can be achieved with geometry b,c or
d (wedge-shaped hoppers).

In some industries non-symmetrical silos are preferred (e.g. pyramidal hoppers
with differently sloped walls). From the view of mass flow design, there
is no reason to build such silos. If mass flow has to be achieved, symmetrical
hoppers usually require the lowest height for the transition from the silo
cross-section to the outlet cross-section to achieve mass flow [10].

6 Application of the results on the design of silos

In section 4 the silo design procedure due to the theory of Jenike [7] was
described in a shortened way. Further details and information can be given
besides the determination of the hopper slope for mass flow and the size
of the outlet to prevent arching. Some examples are listed shortly in the
following (further examples of silo design: [17,18,22,23]):

Details about hopper slope and size of outlet for different hopper forms
(see figure 7) and wall materials. Because of that, a comparison of manufacturing
costs of different hopper forms and hopper materials is possible [18,19].
It can be find out, for example, whether lining of the hopper walls (e.g.
with cold rolled stainless steel sheets) is useful regarding the costs.

If the mass flow design yields an extremely steep mass flow hopper, or if
in the case of a retrofit of an existing silo mass flow should be achieved
without modifying the (too shallow) hopper walls, specially suited installations
can be dimensioned on the basis of the measured flow properties and Jenikes'
theory [15,16].

In case of varying material properties (e.g. moisture [10,20]), it is possible
to find out with shear tests which conditions would yield the worst flow
properties. If the silo is designed for these conditions, proper operation
is ensured in any case.

In case of bulk solids which tend to time consolidation, it can be stated
quantitatively which size of outlet is necessary to avoid arching in dependence
on the storage time at rest. A mass flow silo provides the opportunity to
keep the bulk solid in motion by regular discharge (and recirculation, if
possible) of a small amount of bulk solid. In this way, the time consolidation
and, hence, the size of the outlet [18] can be limited.

With flow property tests (shear tests), the influence of additives (e.g.
flow agents) can be determined in order to find the optimal mixture [13,18].

In case of the storage of bulk solids sensitive to attrition or stresses
as present in a silo, the limit stress can be examined above which that danger
exists [22,23]. Because of those results, the silo can be designed in that
manner that no stresses will occur which would have a negative influence
on the quality of the product.

To avoid vibrations emerging during discharge of a bulk solid, specially-suited
installations can be designed (e.g. discharge tubes) [14,18].

In general, shear tests are also applied for quality control and flowability
tests [13].

7 Summary

The design of silos in order to obtain reliable flow is possible on the basis
of measured material properties and calculation methods. Because badly designed
silos can yield operational problems and a decrease of the product quality,
the geometry of silos should be determined always on the basis of the material
properties. The expenses for testing and silo design are small compared to
the costs of loss of production, quality problems and retrofits.